Analytic torsion for irreducible holomorphic symplectic fourfolds with involution, III: relation with the BCOV invariant

Dai Imaike

Analytic torsion for irreducible holomorphic symplectic fourfolds with involution, III: relation with the BCOV invariant

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Pith's one-line read This paper proves that the BCOV invariant of a Calabi-Yau fourfold obtained from a K3[2]-type manifold with antisymplectic involution equals the author's equivariant analytic torsion up to a constant, and derives Borcherds-product…

desk verdict A technically careful paper that proves a genuine comparison between two analytic torsion invariants and yields explicit Borcherds product formulas for the BCOV invariant of certain Calabi-Yau fourfolds. read the letter →

arxiv 2412.00041 v1 pith:2JTN3TU2 submitted 2024-11-22 math.AG

Dai Imaike This is my paper · ORCID

classification math.AG MSC 14J3214J2858J5214D0711F55

keywords equivariantanalytictorsionBCOVinvariantCalabi-YaufourfoldK3[2]-typemanifoldantisymplecticinvolutionBorcherdsproductperioddomain2-elementaryK3surface

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper tries to prove that two invariants that look unrelated actually coincide on a moduli space: the BCOV invariant of a Calabi-Yau fourfold obtained by resolving the quotient of a hyperkähler fourfold by an antisymplectic involution, and an equivariant analytic torsion invariant of the underlying K3[2]-type manifold with involution. The claimed statement is that, under a rank and irreducibility condition, one is a constant multiple of the other for every 2-elementary K3 surface of a fixed type. The proof compares curvature equations and then uses the boundary of the period domain to force the log-ratio to be constant. In the concrete examples covered, this turns the BCOV invariant into the Petersson norm of a reflective modular form $\Phi$ or $\Phi_k$. A sympathetic reader would care because it connects analytic torsion, mirror-symmetric BCOV invariants, and modular forms on period domains.

What carries the argument

The mechanism is a curvature identity plus a boundary residue computation. For a family of K3[2]-type manifolds with antisymplectic involution, the invariant $\tau_{M,K}$ satisfies $$dd^c\log\tau_{M,K}=\sum_{k=0}^{8}(-1)^k\omega_{H^k}-\frac{\chi}{12}\omega_{WP},$$ and the BCOV invariant of the associated fourfold satisfies the same equation. The proof compares Chern forms of direct image bundles through the blowup isomorphism $H^q(X,\Omega_X^p)^+\oplus H^{q-1}(X^\iota,\Omega_{X^\iota}^{p-1})\cong H^q(Z,\Omega_Z^p)$, together with hyperkähler identities for the $(-1)$-eigenspaces on $H^1(X,\Omega_X^1)$. On the moduli space this makes $\log u$ pluriharmonic away from the discriminant divisor; the boundary law $dd^c\log u=a\,\delta_{\bar D}$ and the residue theorem force $a=0$, so $\log u$ extends over the compactification and is constant.

What would settle it

Compute the quotient $u=\tau_{\mathrm{BCOV}}/\tau_{\widetilde M_0,K}$ along a one-parameter degeneration of 2-elementary K3 surfaces satisfying the hypotheses of Theorem 2.16; the theorem predicts that $\log u$ has zero residue at the discriminant divisor, so $u$ tends to a nonzero constant. Observing a nonzero residue, or a ratio that varies between two points of the moduli space, would refute the comparison.

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Extended reading notes

Core claim

The central claim is Theorem 2.16: if $M_0$ is a primitive hyperbolic 2-elementary sublattice of the K3 lattice with $\operatorname{rk}(M_0)\le 17$ and the roots of $M_0^\perp$ form a single $O(M_0^\perp)$-orbit, then there is a positive constant $C_{M_0}$, depending only on $M_0$, such that for every 2-elementary K3 surface $(Y,\sigma)$ of type $M_0$, $$\tau_{\mathrm{BCOV}}(Z_Y)=C_{M_0}\,\tau_{\widetilde M_0,K}(X_Y,\iota_Y).$$ Here $Z_Y$ is the Calabi-Yau fourfold obtained as the crepant resolution of the quotient of the Hilbert scheme $X_Y=Y^{[2]}$ by the involution induced by $\sigma$. The paper derives this by showing both invariants satisfy the same curvature equation, so their log-ratio is pluriharmonic off the discriminant divisor, and then uses the residue theorem to show the log-ratio extends to a constant on the Baily-Borel compactification.

Load-bearing premise

The proof needs the discriminant divisor $\bar D_{M_0^\perp}$ to be irreducible, so that every intersection of a curve with the divisor gives the same residue; if the divisor has several components, the residue theorem no longer forces the log-ratio to be constant, and the comparison is not derived.

Editorial extensions

If this is right

For any Enriques surface $S$, the BCOV invariant of the universal cover of the Hilbert scheme of two points on $S$ equals a constant multiple of the Petersson norm of the automorphic form $\Phi$.
For each $k=1,\dots,6$, the BCOV invariant of the fourfold built from a 2-elementary K3 surface of type $\Lambda_k(2)^\perp$ equals $C_k\|\Phi_k([Y])\|^{k+1}$.
The comparison shows the ratio $\tau_{\mathrm{BCOV}}/\tau_{\widetilde M_0,K}$ is constant along the whole moduli space, so the BCOV invariant carries no information beyond the equivariant torsion for these families.
These are explicit BCOV formulas on moduli spaces of dimension greater than one, beyond the previously known hypersurface cases.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the discriminant divisor has several irreducible components, the same residue argument would yield one residue per component, so the natural expectation is a piecewise-constant ratio rather than a global constant; this could be checked by computing the ratio near each boundary component.
The equality of curvature equations suggests the two invariants live in the same determinant-line theory, so known birational invariance of the BCOV invariant would imply the paper's Conjecture 2.17 for equivariant torsion whenever a resolution exists.
For any mirror of these fourfolds, the Borcherds-product formulas predict that the genus-one Gromov-Witten series is a power of $\Phi$ or $\Phi_k$, giving a concrete way to search for a mirror.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Hi,

This paper deserves a real referee. The main theorem (2.16) is a genuine new result: under explicit lattice-theoretic hypotheses, the BCOV invariant of the Camere–Garbagnati–Mongardi Calabi-Yau fourfold agrees, up to a constant, with the author's equivariant analytic torsion invariant for the associated K3[2]-type manifold with involution. The applications—expressing the BCOV invariant as Petersson norms of Borcherds products for Enriques surfaces and for the Λ_k(2)^⊥ cases—are new and give the kind of explicit formula that is useful for genus-one mirror symmetry.

The proof is careful. The curvature equation for τ (Theorem 2.4) is checked through a sequence of Chern-class identities; I sampled Lemmas 2.5, 2.7, 2.9, 2.11, and Proposition 2.14, and the algebra is consistent. The overall strategy follows Yoshikawa's method: compare curvature equations, then analyze boundary behavior on the moduli space. The use of the irreducibility of the discriminant divisor to conclude that all boundary residues are equal and therefore vanish is legitimate; without that hypothesis the argument would not close, and the paper says so explicitly.

The main soft spot is the heavy reliance on the author's two earlier preprints, [19] and [20]. Theorems 1.5 and 1.7, which are the backbone of the comparison, are quoted from those papers. That is not a flaw in itself, but it means the correctness of this paper is contingent on results that have not yet appeared in final form. A referee should check those carefully. The second limitation is the irreducibility assumption on the discriminant divisor; it is satisfied in the applications but not derived. The paper is transparent about this, so it is a scope restriction rather than an error.

One more thing: the paper claims to be the first to give explicit BCOV formulas for Calabi-Yau fourfolds with moduli dimension greater than one. That claim is plausible, and the examples indeed have moduli dimensions from 4 to 10, so it holds in this setting.

Overall, this is a solid, honest paper. The dependence on the preprints is a reason to be cautious, but not a reason to desk-reject. Send it to a referee who knows Yoshikawa's work and the BCOV literature.

Referee Report

0 major / 5 minor

Summary. The paper is the third in a series on equivariant analytic torsion for irreducible holomorphic symplectic fourfolds of K3[2]-type with antisymplectic involution. The main result, Theorem 2.16, shows that for a primitive hyperbolic 2-elementary sublattice M0 of the K3 lattice with rk(M0) ≤ 17 and irreducible discriminant divisor \bar D_{M0^\perp}, the BCOV invariant of the Camere–Garbagnati–Mongardi Calabi–Yau fourfold Z_Y agrees with the author's earlier invariant τ_{\tilde M0,K}(X_Y,ι_Y) up to a constant depending only on M0. The proof compares the curvature equations satisfied by the two invariants (Theorem 2.4 and Eq. (2.1)), then analyses the boundary behaviour on the Baily–Borel compactification via residues. Applications in Section 3 express the BCOV invariant as the Petersson norm of the Borcherds Φ-function for the universal cover of the Hilbert scheme of two points on an Enriques surface (Theorem 3.2) and of the Borcherds products Φ_k for 2-elementary K3 surfaces of type Λ_k(2)^\perp (Theorem 3.4).

Significance. If the result stands, this is a substantial contribution: it gives the first comparison between the BCOV invariant of Calabi–Yau fourfolds with moduli dimension greater than one and a torsion-type invariant, and it yields explicit Petersson-norm formulas for those BCOV invariants. The paper has no fitted parameters: the curvature comparison in Theorem 2.4 is a direct identity of characteristic forms, and the constant in Theorem 2.16 is obtained from rigidity of the pluriharmonic extension rather than from normalization. The main limitations are explicitly stated in the manuscript: the comparison is conditional on the irreducibility of \bar D_{M0^\perp}, and several load-bearing ingredients are quoted from the author's companion preprints [19] and [20] (Theorem 1.5, Theorem 1.7, Eq. (2.48), and the current equation (2.50)). These references are specific and the supported steps are identified, and I do not see circularity; independent verification is, however, currently limited by the availability of those preprints.

minor comments (5)

[Abstract and Section 1] There are several typographical errors, including 'in vaiants' in the abstract and 'donoted' in the paragraph preceding Lemma 2.8 in Section 2.3; these should be corrected.
[Section 2.4, Theorem 2.16] The statement uses the notation r(M0) whereas the hypotheses elsewhere use rk(M0); the notation should be unified throughout.
[Section 2.4, Eq. (2.48)] The constant a0 is first introduced in the local degeneration statement (2.48) and then converted to a for the curve coordinate. The phrase 'Replacing t with tε(t)^{1/ν}' is acceptable, but it would help to spell out explicitly that a0 is rational and independent of the chosen degeneration, since that fact is used for the global residue argument.
[Section 3.1 and 3.2] The introduction says the applications hold 'if every small deformation of Z remains a Calabi–Yau fourfold of Camere–Garbagnati–Mongardi.' The proofs of Theorems 3.2 and 3.4 verify this via [7, Theorem 5.1 and A], but the statement of these theorems does not mention this condition; the formulation should be made consistent so that the conditional nature is transparent.
[Section 3.2] The sentence 'At least to the author, it is unclear what is the mirror of the Calabi–Yau 4-folds in Examples 3.1 and 3.3' is a useful honest caveat, but it is placed in the final discussion rather than in the introduction; consider moving or echoing it where the mirror-symmetry context is introduced.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Theorem 2.16 is a genuine comparison of two independently defined invariants, with the proof relying on explicit curvature and boundary computations rather than definitional equivalences or fitted parameters.

full rationale

The paper proves Theorem 2.16 by introducing u = tau_BCOV(Z_Y) * tau_{M0,K}(X_Y, iota_Y)^{-1}. Lemma 2.15 shows that log u is pluriharmonic on the smooth moduli space because both factors satisfy the same curvature equation: the BCOV side by the cited result [9, Proposition 5.10] and the tau side by Theorem 1.5 from the author's previous paper [19]. The remaining step is to upgrade this pluriharmonicity to constancy using boundary asymptotics: (2.49) gives log u = a log|s|^2 + O(log log |s|^{-1}) near the discriminant, obtained from the independent singularity statements [9, Theorem 6.5 and Proposition 6.8] and [20, Theorem 2.12]. The irreducibility assumption on the divisor bar D_{M0^perp} is used explicitly to convert these asymptotics into the global current equation dd^c log u = a delta_{bar D_{M0^perp}} via [20, Lemma 4.2], so that the residue theorem forces a = 0. None of these steps defines tau_{M,K} in terms of tau_BCOV, nor fits any parameter to the target equality. The heavy citation of the author's companion papers [19] and [20] is load-bearing but not circular: those papers establish the curvature equation, invariance, and logarithmic singularity of tau by explicit equivariant analytic torsion computations, and the target relation with the BCOV invariant is not an assumption of any cited result. The theorem is also explicitly conditional on rk(M0) <= 17 and irreducibility, and the applications verify these hypotheses using independent sources. No fitted-input-called-prediction, self-definitional, or author-imported-uniqueness step occurs.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper introduces no free parameters or postulated entities. The constants C_M0, C, C_k are existence constants whose values are not determined; they are not fitted to data. The paper's central claim rests on the author's previously constructed invariant τ (papers I and II), on the BCOV formalism of Eriksson-Freixas i Montplet-Mourougane, and on lattice-theoretic hypotheses on the discriminant divisor.

assumptions (4)

domain assumption The natural Kähler-type chamber K for the sublattice ~M0 exists and satisfies Γ_{~M0⊥,K} = O+(M0⊥), so the moduli space of K3[2]-type manifolds with involution is identified with the moduli space of 2-elementary K3 surfaces.
Invoked in Theorem 1.3, from [19, Theorem 2.28 and Corollary 2.29] and [20, Lemma 4.2]; this identification is used to view τ_{~M0,K} as a function on M_{M0}.
standard math The BCOV invariant and its curvature equation for families of Calabi-Yau fourfolds are taken from Eriksson-Freixas i Montplet-Mourougane [9].
External result used in equation (2.1) and throughout Section 2; specifically [9, Proposition 5.10] gives the curvature equation for τ_BCOV.
domain assumption The lattice-theoretic hypotheses rk(M0) ≤ 17 and Δ(M0^⊥) has a unique O(M0^⊥)-orbit hold for the cases considered.
Stated at the start of Section 2.4 and used throughout the proof of Theorem 2.16; the rank condition ensures the Baily-Borel boundary has codimension at least 2, and the unique orbit gives irreducibility of the discriminant divisor.
domain assumption For the examples, the fixed locus of the antisymplectic involution is either empty (Enriques case) or a union of k rational curves, and the relevant Borcherds products Φ and Φ_k have the stated weight and zero divisor.
Used in Example 3.1 and Example 3.3, citing [26], [28], [7], [5], [6], [33], [16].

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Pith. "Pith review of Analytic torsion for irreducible holomorphic symplectic fourfolds with involution, III: relation with the BCOV invariant." pith.science (2026). https://pith.science/paper/2JTN3TU2

@misc{pith2026241200041,
  author       = {Pith},
  title        = {Pith review of: Analytic torsion for irreducible holomorphic symplectic fourfolds with involution, III: relation with the BCOV invariant},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2JTN3TU2}},
  note         = {Machine review of arXiv:2412.00041}
}

abstract

A Calabi-Yau 4-fold of Camere-Garbagnati-Mongardi is a crepant resolution of the quotient of a hyperk\"ahler 4-fold by an antisymplectic involution. In this paper, we compare two different types of holomorphic torsion invariants; one is the BCOV invariant of the Calabi-Yau 4-fold of Camere-Garbagnati-Mongardi, and the other is the invariant of the corresponding $K3^{[2]}$-type manifold with involution introduced by the author in the preceding papers. As an application, in some special cases, we show that the BCOV invariant of those Calabi-Yau 4-folds is expressed as the Petersson norm of a certain Borcherds product.

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [2]

Bershadsky, S

M. Bershadsky, S. Cecotti, H. Ooguri, and C. Vafa, Kodaira-Spencer theory of gravity and exact re- sults for quantum string amplitudes , Comm. Math. Phys. 165 (1994), no. 2, 311–427. MR 1301851

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Analytic torsion for irreducible holomorphic symplectic fourfolds with involution, I: Construction of an invariant

D. Imaike, Analytic torsion for irreducible holomorphic symplectic f ourfolds with involution, I: Construction of an invariant , preprint arXiv:2406.18023 (2024)

work page Pith review arXiv 2024

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, Analytic torsion for irreducible holomorphic symplectic f ourfolds with involution, II , preprint arXiv:2411.13911 (2024)

work page Pith review arXiv 2024

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Beauville, Antisymplectic involutions of holomorphic symplectic man ifolds, J

A. Beauville, Antisymplectic involutions of holomorphic symplectic man ifolds, J. Topol. 4 (2011), no. 2, 300–304. MR 2805992

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Boissi` ere, C

S. Boissi` ere, C. Camere, and A. Sarti, Classiﬁcation of automorphisms on a deformation family of hyper-K¨ ahler four-folds by p-elementary lattices , Kyoto J. Math. 56 (2016), no. 3, 465–499. MR 3542771

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